1 TRIANGLE OF FORCES LABORATORY - WEEK 7 OR 9 1.1 INTRODUCTION The principle of equilibrium of static forces is useful when considering design of engineering items. As engineers we strive

to ensure that most systems are in equilibrium (i.e., not moving, e.g., sitting on a chair). In its simplest form it can be stated that if three forces acting on a body in a single plane are in equilibrium then their lines of action must meet at a point. The graphical representation of these forces will therefore be a triangle. When a set of forces in random directions in a plane act on a body, and are in equilibrium, it is often convenient to use a graphical solution with forces to a suitable scale. The following lab examines this method and the likely accuracy of using this method compared to the mathematical (trigonometrical) method. Figure 1: Schematic of the experimental setup to demonstrate the equilibrium of static forces principle using a vector triangle with balanced weights. Objectives 1) To resolve, by experiment, any suitable combination of three static, co-planar forces. 2) To compare the results with the graphical solution obtained by drawing a triangle of forces diagram. 3) To illustrate the 'resultant' of two of the forces and to compare the magnitude and direction of its equal and opposite 'equilibrant' with the experimental values. 1.2 MATERIAL AND METHODS 1.2.1 Apparatus Description A circular plate is supported horizontally on three legs (Figure 1). A 360° protractor is fixed centrally on the plate and a steel peg stands up at the middle. There are 3 separate strings that are connected in the centre by a key ring. These strings hold the loads hangers (where the weights are placed). The strings will line up with a line on the protractor - this acts as the direction of the force vector. Three pulleys on clamps enable the load hanger strings to run toward a ring dropped over the peg. A set of weights supplied are placed onto the load hangers to allow loading to be completed (this is the magnitude of the force vector). The weights can be changed, and the string direction can be moved-keep adding/removing weights and moving the strings until the key ring is fully in the centre of the plate (this means the setup is in equilibrium)./n1.2.2 Experimental Procedure 1) Set up three pulley brackets, strings and hangers in any suitable arrangement. Let OP coincide with the zero position of the protractor. 2) Place loads at P and Q. 3) Add a load at R and adjust the load and/or the position of OR until a condition of equilibrium is achieved. It may be helpful to make small changes to the loads at P and Q. Success will be apparent when the ring 'floats' centrally on the centre peg while the apparatus is gently tapped to minimise pulley friction. 4) Record the loads (including the hangers) and their angular directions in table 1. 5) Take image of the three loads, and one image to show balance. (Therefore, 3 images per experiment) ⒸJMulvihill 2 Sept 2023 V_3.2 Create two further different arrangement of the strings and weights (convert to load) to obtain further results. If two of the lines of action are set at right angles to each other this will enable a simple mathematical check to be made on the value of the third force. 1.3 RESULTS The results section should only contain facts and experimental data. There should be no opinion or observations, this is reserved for the conclusions section. Here, you should describe the tables and figures based on what they contain rather than what the data shows. 1.3.1 Experimental Values First, show clear images in one figure of the experimental setup illustrating the 3 different arrangements of forces that are experimentally demonstrating equilibrium. These images should evidence that the key ring is in the centre, detail the angles of each string, and the weigh of each force. Please ensure that these images are all within one figure (see template at the end of this manual for an example)./n1.3.2 Graphical Method By using graphing tools on the computer or by taking clear images of clean and strong images of physical drawings, make a graphical presentation of the experimental setup (Figure 1) for each set of results. First show the 3 forces in the direction they are placed in the experiment, as shown in Figure 2i. Next show your graphical method (if possible) used to add the force vectors (Figure 2ii). Finally, show the resulting triangle that demonstrates equilibrium as shown in Figure 2iii. If the force vectors do not produce a true triangle there is experimental error of directions and/or magnitudes. These errors are acceptable and will happen with low forces, however, you must explain why they happen. Use the vector diagram shown in Figure 2 (iii) to illustrate the magnitude and direction of the resultant are in equilibrium of the two forces first applied. Repeat for 2 more arrangements. (Hint: use different colours for each vector to make it easier to follow). R ⒸJMulvihill Resultant Of P and Q R (i) Figure 2: Illustrations of the graphical method solution used; (i) of the 3 force vectors P, Q, R on a polar system. (ii) the parallelogram law was used to add the force vectors, (iii) the 3 forces forming a closed right angle triangle demonstrating equilibrium. Note: this is representation illustration where no scale is given. 3 Equilibrant Sept 2023 V_3.2 1.3.3 Mathematical Method Next you must now calculate the resultant forces using the mathematical (trigonometrical) method. Again, use equation software (Microsoft Word has an equation tool) or write out your workings clearly and take an image. Ideally the resultant will add to zero as it is in equilibrium./n1.3.3 Mathematical Method Next you must now calculate the resultant forces using the mathematical (trigonometrical) method. Again, use equation software (Microsoft Word has an equation tool) or write out your workings clearly and take an image. Ideally the resultant will add to zero as it is in equilibrium. 1.4 CONCLUSIONS Here you will compare and contrast the results from both the graphical and mathematical method. You will make clear statements about the experiment and refer to figures or results as evidence to back up your statement. Are there differences between the results you calculated or are they the same? If the resultant was not equal to zero was the same value found in both methods? There are certain observations and statements you must definitively address here: While performing the experiment did you see any sources of error? How sensitive was the system in equilibrium to small changes of load or direction of the chords? Generally did the experiment verify the principle of equilibrium? Where could this be applicable in real life? LAB REPORT-EXTRA INFO There is template at the end of this manual with more detailed information on what is required in the lab report. However, it is generic for all 3 labs. Therefore, this subsection will detail more specific information required for this particular lab experiment. Report to be written in a Microsoft Word document. You can draw force diagrams in paint or word or any graphing software you are familiar with. Similar to graphs, an equation software is preferred (i.e. word has an equation editor). You can take a picture of your own drawing of the force diagrams, but you MUST have the scale (e.g. 1mm = 1N) and it MUST be clear to see in the image. You will lose marks if you do not have the following: 1. 3 different arrangements of string directions and forces; 2. each arrangement should have one photo of angles, one photo of loads used, one photo proving equilibrium of central ring, one table of loads and angles (See table), 3 figures of resolved forces (figure 1, 2, and 3, these verify equilibrium). 3. Conclusions (max. 50 words for each arrangement). 4. Remember; this must be repeated 3 times but with different angles of the strings (Conclusion should address each arrangement). 5. Conclusions: include a mention of real world examples of equilibrium where there are multiple forces to be balanced.